Question: A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0).  What is the probability that $x+y < 3$?
Solution: We note that the points for which $x+y<3$ are those that lie below the line $x+y = 3$, or $y= -x + 3$.  As the diagram below illustrates, these are all the points in the square except those in the triangle with vertices (2,1), (2,2), and (1,2).

[asy]
defaultpen(.7);

draw((-.1,0)--(3,0),Arrow);
draw((0,-.1)--(0,4),Arrow);

draw((0,2)--(2,2)--(2,0));
draw((-.5,3.5)--(2.5,.5),dashed,Arrows);

fill((0,0)--(0,2)--(1,2)--(2,1)--(2,0)--cycle,gray(.7));

label("(1,2)",(1,2),NE);
label("(2,2)",(2,2),NE);
label("(2,1)",(2,1),NE);
[/asy]

Since this is a right triangle whose sides both of length 1, its area is $\frac{1}{2} \cdot 1^2 = 1/2$.  Since the square in question has side length 2, its area is $2^2 = 4$, so the shaded region has area $4 - 1/2 = 7/2$.  Our probability is therefore $\dfrac{7/2}{4} = \boxed{\dfrac{7}{8}}$.